DOI QR코드

DOI QR Code

ON THE STABILITY OF THE MIXED TYPE FUNCTIONAL EQUATION IN RANDOM NORMED SPACES VIA FIXED POINT METHOD

  • Jin, Sun-Sook (Department of Mathematics Education, Gongju National University of Education) ;
  • Lee, Yang-Hi (Department of Mathematics Education, Gongju National University of Education)
  • 투고 : 2011.10.25
  • 심사 : 2012.02.20
  • 발행 : 2012.02.28

초록

In this paper, we prove the stability in random normed spaces via fixed point method for the functional equation $$f(x+y+z)-f(x+y)-f(y+z)-f(x+z)+f(x)+f(y)+f(z)=0$$. by using a fixed point theorem in the sense of L. C$\breve{a}$dariu and V. Radu.

키워드

참고문헌

  1. T. Aoki: On the stability of the linear transformation in Banach spaces. J. Math. Soc. Japan 2 (1950), 64-66. https://doi.org/10.2969/jmsj/00210064
  2. P.W. Cholewa: Remarks on the stability of functional equations. Aequationes Math. 27 (1984), 76-86. https://doi.org/10.1007/BF02192660
  3. Z. Gajda: On stability of additive mappings. Internat. J. Math. Math. Sci. 14 (1991), 431-434. https://doi.org/10.1155/S016117129100056X
  4. P. G¸avruta : A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings. J. Math. Anal. Appl. 184 (1994), 431-436. https://doi.org/10.1006/jmaa.1994.1211
  5. D. H. Hyers : On the stability of the linear functional equation. Proc. Natl. Acad. Sci. 27 (1941), 222-224. https://doi.org/10.1073/pnas.27.4.222
  6. S.-S. Jin & Y.-H. Lee: A fixed point approach to the stability of the mixed type functional equation. to appear
  7. S.-M. Jung: On the Hyers-Ulam stability of the functional equations that have the quadratic property. J. Math. Anal. Appl. 222 (1998), 126-137. https://doi.org/10.1006/jmaa.1998.5916
  8. H.-M. Kim: On the stability problem for a mixed type of quartic and quadratic functional equation. J. Math. Anal. Appl. 324 (2006), 358-372. https://doi.org/10.1016/j.jmaa.2005.11.053
  9. Y.-H. Lee: On the stability of the monomial functional equation. Bull. Korean Math. Soc. 45 (2008), 397-403. https://doi.org/10.4134/BKMS.2008.45.2.397
  10. Y.-H. Lee & K.-W. Jun: A generalization of the Hyers-Ulam-Rassias stability of Jensen's equation. J. Math. Anal. Appl. 238 (1999), 305-315. https://doi.org/10.1006/jmaa.1999.6546
  11. Y.-H. Lee & K.-W. Jun: A generalization of the Hyers-Ulam-Rassias stability of Pexider equation. J. Math. Anal. Appl. 246 (2000), 627-638. https://doi.org/10.1006/jmaa.2000.6832
  12. Y.-H. Lee & K.-W. Jun: On the stability of approximately additive mappings. Proc. Amer. Math. Soc. 128 (2000), 1361-1369. https://doi.org/10.1090/S0002-9939-99-05156-4
  13. B. Margolis & J.B. Diaz: A fixed point theorem of the alternative for contractions on a generalized complete metric space. Bull. Amer. Math. Soc. 74 (1968), 305-309. https://doi.org/10.1090/S0002-9904-1968-11933-0
  14. D. Mihet & V. Radu: On the stability of the additive Cauchy functional equation in random normed spaces. J. Math. Anal. Appl. 343 (2008), no. 1, 567-572. https://doi.org/10.1016/j.jmaa.2008.01.100
  15. V. Radu: The fixed point alternative and the stability of functional equations. Sem. Fixed Point Theory 4 (2003), no. 1, 91-96.
  16. Th. M. Rassias : On the stability of the linear mapping in Banach spaces. Proc. Amer. Math. Soc. 72 (1978), 297-300. https://doi.org/10.1090/S0002-9939-1978-0507327-1
  17. I.A. Rus: Principles and applications of fixed point theory. Ed. Dacia, Cluj-Napoca. 1979(in Romanian).
  18. B. Schweizer & A. Sklar: Probabilistic metric spaces. Noth-Holland Series in Probability and Applied Mathematics, North-Holland, New York, NY, USA (1983).
  19. A.N. Serstnev: On the motion of a random normed space. Dokl. Akad. Nauk SSSR 149 (1963), 280-283.
  20. S.M. Ulam: A collection of mathematical problems. Interscience, New York, 1968, p. 63.

피인용 문헌

  1. -Dimensional Quadratic and Additive Type Functional Equation vol.2014, pp.1687-0042, 2014, https://doi.org/10.1155/2014/184680